Is there a relationship between Computer Algebra and NLP?
There isn't, really. Natural language is way more complex and irregular than algebra, which is far more formalised and unambiguous.
So far, in NLP, most success/progress has been made in little toy domains, which exclude most of the complexities of real life, including many ambiguities.
When you say the rules of algebra are somewhat like grammar, then that is because it is essentially a formal language, for which we can specify a grammar. There is currently no complete grammar for any human language (and I doubt there ever will be), let alone a formal one that can be processed by computer.
This was one of the reasons why the first AI boom, where a lot of over-hyped promises where made about being able to translate Russian into English automatically, failed abysmally: natural languages are more than just formal grammars of lexical items.
Stochastic approaches have gone some way towards pragmatic solutions, but when it comes to understanding language they are basically a fudge. And don't get me started on deep learning approaches to NLP.
So the only relationship is that we use the term 'grammar' for the descriptive formalisms in both cases; a formal grammar of algebra would be very different from a grammar for a human language.
This doesn't mean, however, that approaches developed in the field of NLP cannot be applied to algebra: even those which failed in NLP because they were overly limiting. To find out more about this, look for Chomsky Hierarchy -- that describes the different expressive powers of formal languages.
But I would argue that human language is outside of that, because it is not a formal language.